For \(p\in [1,\,\infty ]\) , the notation \(L^{p}(0,\,T;X)\)  stands for the set of measurable functions on the interval \((0,\,T)\)  with values in \(X\)  and \(\Vert f(\cdot ,t)\Vert _{X}\)  belonging to \(L^{p}(0,\,T)\) .
For simplicity, we write \(\Vert f\Vert  _{L^{p,q}(Q(\mu ))}:=\Vert f\Vert  _{L^{p}({black}{-\mu ^{2}},0;L^{q}(B(\mu )))} \)  and
\(\Vert f\Vert  _{L^{p}(Q(\mu ))}:=\Vert f\Vert  _{L^{p}L^{p}(Q(\mu ))}\) . We shall denote by \(\langle f,\,g\rangle \)  the \(L^{2}\)  inner product of \(f\)  and \(g\) .
The classical Sobolev norm \(\Vert \cdot \Vert _{H^{s}}\)  is defined as \(\Vert f\Vert ^{2} _{{H}^{s}}= \int _{\mathbb {R}^{n}} (1+|\xi |)^{2s}|\hat{f}(\xi )|^{2}d\xi \) , \(s\in \mathbb {R}\) .
We denote by \( \dot{H}^{s}\)  homogenous Sobolev spaces with the norm \(\Vert f\Vert ^{2} _{\dot{H}^{s}}= \int _{\mathbb {R}^{n}} |\xi |^{2s}|\hat{f}(\xi )|^{2}d\xi \) . Denote
the average of \(f\)  on the ball \(B(\mu )\)  by
\(\overline{f}_{\mu }\) . \(\Gamma \)  denotes the standard normalized fundamental solution of Laplace equation in \(\mathbb {R}^{3}\) .
We denote by \( the divergence operator in \) R4+\(.\) |S|\( represents the Lebesgue measure of the set \) S\(. We will use the summation convention on repeated indices.\) C\( is an absolute constant which may be different from line to line unless otherwise stated in this paper.\begin{definition}The (upper) box-counting dimension of a set X is usually defined as{black}{\mathrm {dim}_B(X)}=\limsup _{\epsilon \rightarrow 0}{-\log \epsilon },where N(X,\,\epsilon ) is the minimum number of balls of radius \epsilon  required to cover X.\end{definition}{black}{Let \beta >0, \delta >0} and \) I\( can be covered by the {black}{union of series of} parabolic balls \) Q(r)\( with radius \) rj\( less than \)\( for \) jN\(. Define\)\(\mathcal {P}^{\beta }_{\delta }= \inf \Big \lbrace \Sigma r_{j}^{\beta }| \Omega \times I\subseteq \cup Q(r_j), r_{j}<\delta ,j \in \mathbb {N}\Big \rbrace \)\(and \) P=0P\(. If there is \) 0\( such that {black}{\mathcal {P}^{\beta }=\infty } if \) <0\( and {black}{\mathcal {P}^{\beta }=0} if \) >0\({black}{, then} \) 0\( is called as the parabolic Hausdorff dimension and {black}{\mathcal {P}^{\beta }} is the parabolic Hausdorff measure. The details of fractal dimension can be found in \cite {[Falconer]}.\) Next, we focus on Caffarelli and Silvestre's generalized extension for the fractional Laplacian operator \((-\Delta )^{\alpha }\)  with \(0<\alpha <1\)  in [1]}. The fractional power of Laplacian in \(\mathbb {R}^{3}\)  can be interpreted as
\((-\Delta )^{\alpha }u=-C_{\alpha }\lim \limits _{y\rightarrow 0_{+}}y^{1-2\alpha }\partial _{y}u^{\ast },\) 
